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An [[Affine connection|affine connection]] on a smooth manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091810/s0918101.png" /> of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091810/s0918102.png" /> with a non-degenerate <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091810/s0918103.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091810/s0918104.png" /> that is covariantly constant with respect to it. If the affine connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091810/s0918105.png" /> is given by the local connection forms
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091810/s0918106.png" /></td> </tr></table>
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091810/s0918107.png" /></td> </tr></table>
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An [[Affine connection|affine connection]] on a smooth manifold  $  M $
 +
of dimension  $  2n $
 +
with a non-degenerate  $  2 $-form  $  \Phi $
 +
that is covariantly constant with respect to it. If the affine connection on  $  M $
 +
is given by the local connection forms
 +
 
 +
$$
 +
\omega  ^ {i}  = \
 +
\Gamma _ {k}  ^ {i}  dx  ^ {k} ,\ \
 +
\mathop{\rm det}  \| \Gamma _ {k}  ^ {i} \|  \neq  0,
 +
$$
 +
 
 +
$$
 +
\omega _ {j}  ^ {i}  = \Gamma _ {jk}  ^ {i} \omega  ^ {k}
 +
$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091810/s0918108.png" /></td> </tr></table>
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$$
 +
\Phi  = \
 +
a _ {ij} \omega  ^ {i} \wedge \omega  ^ {j} ,\ \
 +
a _ {ij}  = - a _ {ji} ,
 +
$$
  
then the condition that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091810/s0918109.png" /> be covariantly constant can be expressed in the form
+
then the condition that $  \Phi $
 +
be covariantly constant can be expressed in the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091810/s09181010.png" /></td> </tr></table>
+
$$
 +
da _ {ij}  = \
 +
a _ {kj} \omega _ {i}  ^ {k} +
 +
a _ {ik} \omega _ {j}  ^ {k} .
 +
$$
  
The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091810/s09181011.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091810/s09181012.png" /> defines a symplectic (or almost-Hamiltonian) structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091810/s09181013.png" /> that converts every tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091810/s09181014.png" /> into a symplectic space with the skew-symmetric scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091810/s09181015.png" />. A symplectic connection can also be defined as an affine connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091810/s09181016.png" /> which preserves this product under parallel transfer of vectors. In every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091810/s09181017.png" /> one can choose a frame such that
+
The $  2 $-form $  \Phi $
 +
defines a symplectic (or almost-Hamiltonian) structure on $  M $
 +
that converts every tangent space $  T _ {x} ( M) $
 +
into a symplectic space with the skew-symmetric scalar product $  \Phi ( X, Y) $.  
 +
A symplectic connection can also be defined as an affine connection on $  M $
 +
which preserves this product under parallel transfer of vectors. In every $  T _ {x} ( M) $
 +
one can choose a frame such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091810/s09181018.png" /></td> </tr></table>
+
$$
 +
\Phi  = 2
 +
\sum _ {\alpha = 1 } ^ { n }
 +
\omega  ^  \alpha  \wedge
 +
\omega ^ {n + \alpha } .
 +
$$
  
The set of all such frames forms a principal fibre bundle over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091810/s09181019.png" />, whose structure group is the [[Symplectic group|symplectic group]]. A symplectic connection is just a connection in this principal fibre bundle. There are manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091810/s09181020.png" /> of even dimension on which there is no non-degenerate globally defined <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091810/s09181021.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091810/s09181022.png" /> and, consequently, no symplectic connection. However, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091810/s09181023.png" /> exists, a symplectic connection with respect to which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s091/s091810/s09181024.png" /> is covariantly constant is not uniquely determined.
+
The set of all such frames forms a principal fibre bundle over $  M $,  
 +
whose structure group is the [[Symplectic group|symplectic group]]. A symplectic connection is just a connection in this principal fibre bundle. There are manifolds $  M $
 +
of even dimension on which there is no non-degenerate globally defined $  2 $-form $  \Phi $
 +
and, consequently, no symplectic connection. However, if $  \Phi $
 +
exists, a symplectic connection with respect to which $  \Phi $
 +
is covariantly constant is not uniquely determined.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Sternberg,  "Lectures on differential geometry" , Prentice-Hall  (1964)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Sternberg,  "Lectures on differential geometry" , Prentice-Hall  (1964)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Abraham,  J.E. Marsden,  "Foundations of mechanics" , Benjamin/Cummings  (1978)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  R. Abraham,  J.E. Marsden,  "Foundations of mechanics" , Benjamin/Cummings  (1978)</TD></TR></table>

Latest revision as of 02:36, 14 September 2022


An affine connection on a smooth manifold $ M $ of dimension $ 2n $ with a non-degenerate $ 2 $-form $ \Phi $ that is covariantly constant with respect to it. If the affine connection on $ M $ is given by the local connection forms

$$ \omega ^ {i} = \ \Gamma _ {k} ^ {i} dx ^ {k} ,\ \ \mathop{\rm det} \| \Gamma _ {k} ^ {i} \| \neq 0, $$

$$ \omega _ {j} ^ {i} = \Gamma _ {jk} ^ {i} \omega ^ {k} $$

and

$$ \Phi = \ a _ {ij} \omega ^ {i} \wedge \omega ^ {j} ,\ \ a _ {ij} = - a _ {ji} , $$

then the condition that $ \Phi $ be covariantly constant can be expressed in the form

$$ da _ {ij} = \ a _ {kj} \omega _ {i} ^ {k} + a _ {ik} \omega _ {j} ^ {k} . $$

The $ 2 $-form $ \Phi $ defines a symplectic (or almost-Hamiltonian) structure on $ M $ that converts every tangent space $ T _ {x} ( M) $ into a symplectic space with the skew-symmetric scalar product $ \Phi ( X, Y) $. A symplectic connection can also be defined as an affine connection on $ M $ which preserves this product under parallel transfer of vectors. In every $ T _ {x} ( M) $ one can choose a frame such that

$$ \Phi = 2 \sum _ {\alpha = 1 } ^ { n } \omega ^ \alpha \wedge \omega ^ {n + \alpha } . $$

The set of all such frames forms a principal fibre bundle over $ M $, whose structure group is the symplectic group. A symplectic connection is just a connection in this principal fibre bundle. There are manifolds $ M $ of even dimension on which there is no non-degenerate globally defined $ 2 $-form $ \Phi $ and, consequently, no symplectic connection. However, if $ \Phi $ exists, a symplectic connection with respect to which $ \Phi $ is covariantly constant is not uniquely determined.

References

[1] S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964)

Comments

References

[a1] R. Abraham, J.E. Marsden, "Foundations of mechanics" , Benjamin/Cummings (1978)
How to Cite This Entry:
Symplectic connection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Symplectic_connection&oldid=11792
This article was adapted from an original article by Ü. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article